Subelliptic Estimates and Finite Type

Summary

This paper surveys work in partial differential equations and several complex variables that revolves around subelliptic estimates in the ∂-Neumann problem. The paper begins with a discussion of the question of local regularity; one is given a bounded pseudoconvex domain with smooth boundary, and hopes to solve the inhomogeneous system of Cauchy- Riemann equation ∂u = α, where a is a differential form with square integrable coefficients and satisfying necessary compatibility conditions. Can one find a solution u that is smooth wherever α is smooth? According to a fundamental result of Kohn and Nirenberg, the answer is yes when there is a subelliptic estimate. The paper sketches the proof of this result, and goes on to discuss the history of various finite-type conditions on the boundary and their relationships to subelliptic estimates. This includes finite-type conditions involving iterated commutators of vector fields, subelliptic multipliers, finite type conditions measuring the order of contact of complex analytic varieties with the boundary, and Catlin's multitype. The paper also discusses additional topics such as nonpseudoconvex domains, Holder and L^p estimates for ∂, and finite-type conditions that arise when studying holomorphic extension, convexity, and the Bergman kernel function. The paper contains a few new examples and some new calculations on CR manifolds. The paper ends with a list of nine open problems.